A pair of functions 
and 
are orthonormal if they are orthogonal and each normalized so that
![int_a^b[phi_i(x)]^2w(x)dx](https://mathworld.wolfram.com/images/equations/OrthonormalFunctions/Inline3.svg) |  |  | (1) |
![int_a^b[phi_j(x)]^2w(x)dx](https://mathworld.wolfram.com/images/equations/OrthonormalFunctions/Inline6.svg) |  |  | (2) |
These two conditions can be succinctly written as
 | (3) |
where 
is a weighting function and 
is the Kronecker delta.
Orthonormal Functions
See also
Complete Biorthogonal System, Complete Orthogonal System, Orthogonal Polynomials, Orthonormal Basis, Orthonormal VectorsExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Orthonormal Functions." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OrthonormalFunctions.html